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x^{2}+4x+15=0
All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.
x=\frac{-4±\sqrt{4^{2}-4\times 15}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 4 for b, and 15 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-4±\sqrt{16-4\times 15}}{2}
Square 4.
x=\frac{-4±\sqrt{16-60}}{2}
Multiply -4 times 15.
x=\frac{-4±\sqrt{-44}}{2}
Add 16 to -60.
x=\frac{-4±2\sqrt{11}i}{2}
Take the square root of -44.
x=\frac{-4+2\sqrt{11}i}{2}
Now solve the equation x=\frac{-4±2\sqrt{11}i}{2} when ± is plus. Add -4 to 2i\sqrt{11}.
x=-2+\sqrt{11}i
Divide -4+2i\sqrt{11} by 2.
x=\frac{-2\sqrt{11}i-4}{2}
Now solve the equation x=\frac{-4±2\sqrt{11}i}{2} when ± is minus. Subtract 2i\sqrt{11} from -4.
x=-\sqrt{11}i-2
Divide -4-2i\sqrt{11} by 2.
x=-2+\sqrt{11}i x=-\sqrt{11}i-2
The equation is now solved.
x^{2}+4x+15=0
Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.
x^{2}+4x+15-15=-15
Subtract 15 from both sides of the equation.
x^{2}+4x=-15
Subtracting 15 from itself leaves 0.
x^{2}+4x+2^{2}=-15+2^{2}
Divide 4, the coefficient of the x term, by 2 to get 2. Then add the square of 2 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+4x+4=-15+4
Square 2.
x^{2}+4x+4=-11
Add -15 to 4.
\left(x+2\right)^{2}=-11
Factor x^{2}+4x+4. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.
\sqrt{\left(x+2\right)^{2}}=\sqrt{-11}
Take the square root of both sides of the equation.
x+2=\sqrt{11}i x+2=-\sqrt{11}i
Simplify.
x=-2+\sqrt{11}i x=-\sqrt{11}i-2
Subtract 2 from both sides of the equation.
x ^ 2 +4x +15 = 0
Quadratic equations such as this one can be solved by a new direct factoring method that does not require guess work. To use the direct factoring method, the equation must be in the form x^2+Bx+C=0.
r + s = -4 rs = 15
Let r and s be the factors for the quadratic equation such that x^2+Bx+C=(x−r)(x−s) where sum of factors (r+s)=−B and the product of factors rs = C
r = -2 - u s = -2 + u
Two numbers r and s sum up to -4 exactly when the average of the two numbers is \frac{1}{2}*-4 = -2. You can also see that the midpoint of r and s corresponds to the axis of symmetry of the parabola represented by the quadratic equation y=x^2+Bx+C. The values of r and s are equidistant from the center by an unknown quantity u. Express r and s with respect to variable u. <div style='padding: 8px'><img src='https://opalmath.azureedge.net/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>
(-2 - u) (-2 + u) = 15
To solve for unknown quantity u, substitute these in the product equation rs = 15
4 - u^2 = 15
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = 15-4 = 11
Simplify the expression by subtracting 4 on both sides
u^2 = -11 u = \pm\sqrt{-11} = \pm \sqrt{11}i
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =-2 - \sqrt{11}i s = -2 + \sqrt{11}i
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.